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Resolvent estimates for elliptic finite element operators in one dimension

Authors: M. Crouzeix, S. Larsson and V. Thomée
Journal: Math. Comp. 63 (1994), 121-140
MSC: Primary 65N30; Secondary 65M60
MathSciNet review: 1242058
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Abstract: We prove the analyticity (uniform in h) of the semigroups generated on $ {L_p}(0,1),1 \leq p \leq \infty $, by finite element analogues $ {A_h}$ of a one-dimensional second-order elliptic operator A under Dirichlet boundary conditions. This is accomplished by showing the appropriate estimates for the resolvents by means of energy arguments. The results are applied to prove stability and optimal-order error bounds for numerical solutions of the associated parabolic problem for both smooth and nonsmooth data.

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Keywords: Resolvent, sectorial, elliptic, analytic semigroup, Banach space, $ {L_p}$, maximum norm, finite element method, rational approximation, stability, error estimate
Article copyright: © Copyright 1994 American Mathematical Society

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